On Connected Graphs with Integer-Valued Q-Spectral Radius

Document Type : Original paper

Authors

Department of Mathematical Sciences, Indian Institute of Technology (Banaras Hindu University), Varanasi-221005, India

Abstract

The $Q$-eigenvalues are the eigenvalues of the signless Laplacian matrix $Q(G)$ of a graph $G$, and the largest $Q$-eigenvalue is known as the $Q$-spectral radius $q(G)$ of $G$. The edge-degree of an edge is defined as the number of edges adjacent to it. In this article, we characterize the structure of simple connected graphs having integral $Q$-spectral radius. We show that the necessary and sufficient condition for such graphs to contain either a double star $\mathcal{S}_{r}^{2}$ or its variation $\mathcal{S}_{r}^{2,1}$ (having exactly one common neighbor between the central vertices) as a subgraph is that the maximum edge-degree is $2r$, where $r= q(G) -3$. In particular, we characterize all graphs that contain only double star as a subgraph when $q(G)$ equals $8$ and $9$. Further, we characterize all the connected edge-non-regular graphs with a maximum edge-degree equal to $4$ whose minimum  $Q$-eigenvalue does not belong to the open interval $(0,1)$ and has an integral $Q$-spectral radius.

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References

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Communications in Combinatorics and Optimization
Volume 11, Issue 3
September 2026
Pages 767-786

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